Finite Group Databasebeta

← #115162

Subgroups — #115162

Order 512.

1084 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2399C211yesmaximal, characteristic
2#21822C211yesmaximal, characteristic
3C2×C4×#101C211yesmaximal, characteristic
4C2×#1491C212yesmaximal
5C2×#1300C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C2×#432C2212yes—
9C22×#101—21no—
10C2×C4×M16C2212yes—
11#2399—24no—
12#2399—24no—
13C2×#398—21no—
14C2×#398C412yes—
15C2×#395—21no—
16C2×#395C412yes—
17C2×C4×C4⋊C4—22no—
18C2×#376C2211yescharacteristic
19#1985—24no—
20#1817—22no—
21C4×#101—24no—
22C2×#343—21no—
23#1491—24no—
24#1491—24no—
25C2×Q8⋊C8—24no—
26#1300—24no—
27C23×Q8C2×C411yescharacteristic
28C23×Q8D411yescharacteristic
29C22×M16—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8D414yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C42—21no—
49C22×C42—21no—
50C22×C42D412yes—
51C22×C42C2311yescharacteristic
52#432—22no—
53#432—22no—
54#432—22no—
55#432—22no—
56C2×C4⋊C8D414yes—
57C2×#101—28no—
58C2×#101—24no—
59C2×C22⋊C8—22no—
60C4×M16—22no—
61C4×M16—22no—
62C4×M16—22no—
63C4×M16D414yes—
64C2×#97—22no—
65C2×#97—22no—
66C2×C4×C8—22no—
67#398—28no—
68#398—24no—
69#398—24no—
70#395—28no—
71#395—24no—
72#395—24no—
73C4×C4⋊C4—216no—
74#376—22no—
75#376D414yes—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95C2×C412yes—
80#343—24no—
81#336—28no—
82Q8⋊C8—216no—
83C22×Q8—24no—
84C22×Q8—21no—
85C22×Q8—21no—
86C22×Q8—22no—
87C22×Q8—22no—
88C22×Q8—21no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—42no—
92C22×Q8—44no—
93C22×Q8C22⋊C412yes—
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C2×M16—28no—
103C2×M16—24no—
104C2×M16—44no—
105C2×M16—42no—
106C22×C8—24no—
107C22×C8—42no—
108C4×Q8—24no—
109C4×Q8—24no—
110C4×Q8—24no—
111C4×Q8—24no—
112C4×Q8—42no—
113C4×Q8—42no—
114C4×Q8—42no—
115C4×Q8—42no—
116C4×Q8—42no—
117C4×Q8—42no—
118C4×Q8—42no—
119C4×Q8—42no—
120C4×Q8—42no—
121C4×Q8—42no—
122C4×Q8—42no—
123C4×Q8—42no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—48no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4C22⋊C412yes—
145C2×C42—28no—
146C2×C42—24no—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—22no—
150C2×C42—24no—
151C2×C42—22no—
152C2×C42—41no—
153C2×C42—42no—
154C2×C42—41no—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159C2×C42#4112yes—
160C2×C42#4112yes—
161C4⋊C8—24no—
162C4⋊C8—24no—
163#101—216no—
164C22⋊C8—42no—
165C22⋊C8—42no—
166#97—24no—
167#97—42no—
168#97—42no—
169#97—42no—
170C4×C8—24no—
171C4×C8—42no—
172#95—28no—
173#95—28no—
174#95—22no—
175#95—24no—
176#95C22⋊C414yes—
177C24C2×C22⋊C411yescharacteristic
178C2×Q8—24no—
179C2×Q8—24no—
180C2×Q8—28no—
181C2×Q8—24no—
182C2×Q8—48no—
183C2×Q8—48no—
184C2×Q8—44no—
185C2×Q8—44no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—44no—
189C2×Q8—48no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—28no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4#10414yes—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C4×D412yes—
226C22×C4C4×D412yes—
227C22×C4#12011yescharacteristic
228C22×C4#12012yes—
229C22×C4#12012yes—
230C22×C4#12011yescharacteristic
231C22×C4#12112yes—
232C22×C4#12112yes—
233C22×C4#12311yescharacteristic
234C22×C4#12311yescharacteristic
235M16—28no—
236M16—44no—
237M16—48no—
238C2×C8—48no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—48no—
242C4⋊C4—24no—
243C4⋊C4—416no—
244C4⋊C4—416no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C42—216no—
253C42—44no—
254C42—44no—
255C42—44no—
256C42—44no—
257C42#12012yes—
258C42#12112yes—
259C42#12112yes—
260C42#12112yes—
261C42#12112yes—
262C42#12312yes—
263C42#12412yes—
264C42#12712yes—
265C23—22no—
266C23—22no—
267C23#37911yescharacteristic
268C23#38611yescharacteristic
269C23#38611yescharacteristic
270C23C2×#10412yes—
271C23#42112yes—
272Q8—28no—
273Q8—48no—
274Q8—48no—
275Q8—44no—
276Q8—48no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—28no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—82no—
316C2×C4#38612yes—
317C2×C4#39012yes—
318C2×C4#39212yes—
319C2×C4#39212yes—
320C2×C4#39412yes—
321C2×C4#39412yes—
322C2×C4#45614yes—
323C8—48no—
324C8—84no—
325C22—24no—
326C22—24no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22#143811yescharacteristic
332C22#178912yes—
333C22#187112yes—
334C22#188612yes—
335C22#189112yes—
336C22#189611yescharacteristic
337C22#189711yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—48no—
344C4—44no—
345C4—44no—
346C4—82no—
347C4—82no—
348C4#200514yes—
349C2—22no—
350C2—21no—
351C2—21no—
352C2#903014yes—
353C2#1097111yescharacteristic
354C2#1097811yescharacteristic
355C2#1099311yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#23992
#218222
C2×C4×#1012
C2×#14912
C2×#13002